Self‐Healing Ability of Perovskites Observed via Photoluminescence Response on Nanoscale Local Forces and Mechanical Damage

Abstract The photoluminescence (PL) of metal halide perovskites can recover after light or current‐induced degradation. This self‐healing ability is tested by acting mechanically on MAPbI3 polycrystalline microcrystals by an atomic force microscope tip (applying force, scratching, and cutting) while monitoring the PL. Although strain and crystal damage induce strong PL quenching, the initial balance between radiative and nonradiative processes in the microcrystals is restored within a few minutes. The stepwise quenching–recovery cycles induced by the mechanical action is interpreted as a modulation of the PL blinking behavior. This study proposes that the dynamic equilibrium between active and inactive states of the metastable nonradiative recombination centers causing blinking is perturbed by strain. Reversible stochastic transformation of several nonradiative centers per microcrystal under application/release of the local stress can lead to the observed PL quenching and recovery. Fitting the experimental PL trajectories by a phenomenological model based on viscoelasticity provides a characteristic time of strain relaxation in MAPbI3 on the order of 10–100 s. The key role of metastable defect states in nonradiative losses and in the self‐healing properties of perovskites is suggested.

To estimate the effective pressure from the applied force, we have employed the Johnson-Kendall-Roberts (JKR) model, [1] which is considered to be applicable for soft samples with large adhesion forces between tips and samples and large tip sizes. [2] Since no large tips were used, our treatment is only a rough estimate. The reduced Young's modulus was calculated using the elastic modulus E = 157.5 GPa and a Poisson's ratio ν = 0.27 for silicon, [2] as well as E = 13.9 GPa and ν = 0.33 for MaPbI3. [3] As tip radius, 7 nm (given by the manufacturer) was used. The adhesion force was roughly estimated from the experimentally observed, average pull-off-forces of the force-distancecurves to be ~10 nN. Based on these assumptions, contact radii of 2.4-4.9 nm and contact areas of

Supplementary Note 2. Theoretical model based on visco-elasticity
The basic assumption of the model is that PL intensity directly depends on crystal strain . The time dependence of applied stress in the experiment when force is applied at time 0 and released at time toff is given by Assuming that the perovskite nanocrystal can be described by the Voigt model [4] one can write the equation for the strain time dependence ( ) where is elasticity coefficient and is the viscosity. Solving Eq.S5 with the stress dependence Eq.S4 we have: where is relaxation time of strain: We assume that the PL intensity directly depends on the strain: where the function ( ) must be found from the experimental data.
Let us fit the experimental intensity as a function of theoretical ( ) given by (S9) for both PL decline (when the force is applied) and PL recovery when the force is removed with the same fitting parameter . It turned out that indeed our experimental data can be fitted in this way (see Figure 9 in the main text).
The obtained L( ) dependence can be approximated by the following function: Thus, the time dependence ( ) can be now approximated by the formula were ( ) is given by Eq.S9. The results of the fitting for two different crystals are shown in the main text in Figure 9.

Supplementary Note 3. PL decline time and half-recovery time in the framework of the theoretical model based on visco-elasticity
PL decline time is determined as the time passed from the moment of force application until the PL intensity reaches a stable low intensity level. It means that within this time the first term in Eq.S11 becomes much smaller (for example 20 times less) than its initial value at = 0. We can write this condition as: The dependence ( ) given by Eq.S6 can be approximated for small by a linear function: From the last two equations we obtain: (S12) So, the PL decline time is inversely proportional to the applied force.
Half-recovery time is defined as the time interval from off until the moment when PL recovers to 50% of its initial value. Mathematically we can write this condition as: or ( off + ) = 0 ln (2 + 1 ) By substituting this to Eq.S6 we obtain the expression for the half-recovery time: So, in the model based on viscoelasticity is very weakly (logarithmically) dependent on the applied force which agrees with the experimental observations.

Supplementary Note 4. A possible model for the switching of the NR centres, which can explain the experimental data.
In this model we realize the idea presented in the paper that the switching dynamics of the metastable NR centres is affected by the strain and this effect only causes reversible PL quenching.
Supplementary Figure 5. Configuration space of a metastable non-radiative center. One of many possible models. PL is quenched by the center when it is in active state.
In the presence of N non-radiative centres, the PL intensity of a semiconductor crystal is determined by stochastic dynamics of the variables ( ), = 1 … , each of them can abruptly Energy Configuration coordinate of the supertrap change its value from 0 (inactive state of the centre, no NR recombination occurs) to 1 (active state of the centre, NR recombination rate is equal to ). Let us assume the simplest dependence of the PL intensity on the activity of the centres (Supplementary Figure 5): where Φ is the emission intensity from the rest of the crystal not being affected by stress, Φ 0 + Φ is the emission intensity when all metastable centers are inactive and is ratio between the nonradiative rate of the recombination by one NR center and the radiative recombination rate.
Let us assume that for each NR centre the rates of switching from 0 to 1 (γ 0→1 ) and from 1 to 0 (γ 1→0 ) are determined by thermal activation: Where the hight of the barrier 1 → 0 for the centre i and Δ is the energy difference between the configuration states 1 and 0, see Supplementary Figure 5. A broad distribution of gives the multi-timescale character of the PL blinking dynamics. Averaged value of over the switching dynamics is: Let us assume that the energy difference between state 1 and state 0 for each NR centre depends linearly on strain (and thus on time in our experiments with temporal applying of local force) and this dependence is the same for all centres: Where Δ 0 -energy difference for center i at zero strain. The dependence ( ) is given by eq. (S6). Substituting in Eq. (S14) each ( ) by 〈 〉, we obtain: This is not an exact equality, it is an approximation, because the used procedure is not a formally exact averaging. After some re-arrangement we obtain: 〈 ( )〉 ≈ Φ 0 + 1 1 + ( + 1) exp (− Δ 0 ) exp( ( )/ 0 ) we find a dependence which is analogues to Eq. (S11): 〈 ( )〉 ≈ 1+ exp( ( )/ 0 ) + 0 (S15) So, in the framework of this particular model, when we average over the fast blinking dynamics, we obtain the same dependence of the PL over time Eq. (S15) as in the pure phenomenological theory based on visco-elasticity (Eq.(S11)).
Supplementary Note 5. Sample identification numbers: